exact soliton solutions of the one-dimensional complex ...people.physics.anu.edu.au/~nna124/physica...

Physica D 176 (2003) 44–66

Exact soliton solutions of the one-dimensionalcomplex Swift–Hohenberg equation

Ken-ichi Marunoa,b,∗, Adrian Ankiewiczc, Nail Akhmedievba Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka 816-8580, Japan

b Optical Sciences Centre, Research School of Physical Sciences and Engineering,The Australian National University, Canberra ACT 0200, Australia

c Applied Photonics Group, Research School of Physical Sciences and Engineering,The Australian National University, Canberra ACT 0200, Australia

Received 9 May 2002; received in revised form 1 October 2002; accepted 2 October 2002Communicated by I. Gabitov

Abstract

Using Painlevé analysis, the Hirota multi-linear method and a direct ansatz technique, we study analytic solutions of the(1 + 1)-dimensional complex cubic and quintic Swift–Hohenberg equations. We consider both standard and generalizedversions of these equations. We have found that a number of exact solutions exist to each of these equations, provided that thecoefficients are constrained by certain relations. The set of solutions include particular types of solitary wave solutions, hole(dark soliton) solutions and periodic solutions in terms of elliptic Jacobi functions and the Weierstrass℘ function. Althoughthese solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic andquintic Swift–Hohenberg equations, those presented here are the first examples of exact analytic solutions found thus far.© 2002 Elsevier Science B.V. All rights reserved.

PACS: 02.30.Jr; 05.45.Yv; 42.65.Sf; 42.65.Tg; 42.81.Dp; 89.75.Fb

Keywords: Solitons; Singularity analysis; Hirota multi-linear method; Complex Swift–Hohenberg equation; Direct ansatz method

1. Introduction

Complicated pattern-forming dissipative systems can be described by the Swift–Hohenberg (S–H) equation[1,2].Classic examples are the Rayleigh–Bénard problem of convection in a horizontal fluid layer in the gravitationalfield [3,4], Taylor–Couette flow[5], some chemical reactions[6] and large-scale flows and spiral core instabilities[7]. Examples in optics include synchronously pumped optical parametric oscillators[8], three-level broad-areacascade lasers[9] and large aspect-ratio lasers[10–13]. In the context of large-aperture lasers with small detuningbetween the atomic and cavity frequencies, the complex cubic Swift–Hohenberg (CCSH) equation was derived

∗ Corresponding author. Present address: Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka 816-8580, Japan.Tel.: +81-925-837684; fax:+81-925-751159.E-mail address: [email protected] (K.-i. Maruno).

0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0167-2789(02)00708-X

K. Maruno et al. / Physica D 176 (2003) 44–66 45

asymptotically[14–16]. This equation is also believed to be relevant for oscillatory convection in binary fluids. Theappearance of spatial patterns is the most remarkable feature of the solutions.

Despite their visual complexity, spatial patterns are actually formed from a certain number of ordered or chaoticcombinations in space of some simpler localized structures. It is known that the S–H equation admits the existenceof transverse localized structures and phase domains[14]. These coherent structures can be considered as bright ordark solitons. Consequently, these simple localized structures and their stability are of great interest in the studyof any pattern-forming system. If we keep these visual properties of patterns in mind, we can split the probleminto several steps. Firstly, we should study the simplest stationary objects (localized structures). Secondly, wehave to investigate their stability. As a last step, we should study their interaction and the combination structures.The distinction between stationary and moving patterns is defined by the nature of the interaction forces betweenthe localized structures. The above program cannot be carried out in one move. Moreover, in studying stationarylocalized structures, we have to start with the simplest cases, viz (1+ 1)-dimensional structures. This sequentialstep-by-step approach allows us to avoid any possible confusion inherent in trying to explain a complicated structurein one step.

Whichever part of the above program we consider, it is clear that it can mainly be done only using computersimulations. This has been done in the majority of papers published so far. We will analyze only (1+1)-dimensionalbright and dark solitons, and leave aside, for the moment, the question of their stability and the interaction betweenthem. Moreover, our task here is to obtain some analytic results. The latter point will further restrict the class ofsolutions which are of interest in our investigation. Nonetheless, this is a step of paramount importance because,until now, no analytic solutions have been known.

In many respects, localized structures of the S–H equation are similar to those observed in systems describedby the previously studied complex Ginzburg–Landau equation (CGLE). We recall that some analytic solutionsfor the cubic and quintic CGLE are known[17–20]. For the cubic (1+ 1)-dimensional equation, analytic solu-tions describe all possible bright and dark soliton solutions. In contrast, in the case of the quintic equation, theanalytic solutions of the CGLE represent only a small subclass of its soliton solutions. Moreover, the stable soli-ton solutions are located outside of this subclass[20]. Therefore, practically useful results can only be obtainednumerically.

We note that, apart from some exceptions, the CGLE generally has only isolated solutions[20,21], i.e. theyare fixed for any particular set of equation parameters. This property is fundamental for the whole set of lo-calized solutions of the CGLE. The existence of isolated solutions is one of the basic features of dissipativesystems in general. The qualitative physical foundations of this property are given in[22]. This property is oneof the reasons why the above program is possible at all. Like the CGLE, the S–H equation models dissipativesystems, and we expect that it will have this property. Indeed, preliminary numerical simulations support thisconjecture.

The main difference between the S–H equation[23] and the CGLE lies in its more involved diffraction term. Thelatter is important in describing more detailed features of an actual physical problem. However, these complicationsprevent us from easily analyzing the solutions. In fact, it was not clear that such solutions could exist at all[8,11]. Inthis work, we study the quintic complex S–H equation in one-dimensional and report various new exact solutions.

Before going into further details, we should distinguish between the real Swift–Hohenberg (RSH) equation andthe CCSH equation. The former can be considered as a particular case of the latter. Moreover, the RSH equationis a phenomenological model which cannot be rigorously derived from the original equations[3]. In contrast, theCCSH equation is derived asymptotically and is rigorous in the long-wavelength limit. In standard notations, theCCSH equation can be written as

ψt = rψ − (1 + ic)|ψ |2ψ + iaψ − d(Λ+)2ψ, (1)

46 K. Maruno et al. / Physica D 176 (2003) 44–66

wherer is the control parameter anda characterizes the diffraction properties of the active medium. In the limit ofΛ → 0, i.e., in the long-wavelength limit, the differential nonlinearities formally have higher orders and thereforecan be dropped. In this case, the wavenumber selecting term,(Λ+)2ψ , is just a small correction to the diffractionterm iaψ . Thus the CCSH equation can be treated as a perturbed cubic CGLE. For us the main interest here is(1 + 1)-dimensional case when has onlyx-derivatives.

In physical problems, the quintic nonlinearity can be of equal or even higher importance to the cubic one[24]as it is responsible for stability of localized solutions. Sakaguchi and Brand made a numerical investigation of thecomplex quintic Swift–Hohenberg (CQSH) equation

ψt = aψ + b|ψ |2ψ − c|ψ |4ψ − d(1 + ∂xx)2ψ + if ∂xxψ, (2)

where all coefficientsa = a1 + ia2, b = b1 + ib2, c = c1 + ic2, and the order parameterψ are complex, butd andf are real[23]. Fora, b andd real andc ≡ 0 andf ≡ 0, the original S–H equation, which was derived as an orderparameter equation for the onset of Rayleigh–Bénard convection in a simple fluid, is recovered. ThenEq. (2)canbe rewritten as

iψt + (f + 2id)ψxx + idψxxxx + (b2 − ib1)|ψ |2ψ + (−c2 + ic1)|ψ |4ψ = (−a2 + i(a1 − d))ψ. (3)

This equation can be generalized to

iψt + βψxx + γψxxxx + µ|ψ |2ψ + ν|ψ |4ψ = iδψ, (4)

where all coefficientsβ = β1 + iβ2, γ = γ1 + iγ2, µ = µ1 + iµ2 andν = ν1 + iν2 are complex andδ is real.Eq. (4)is written in the form which we will refer to as the generalized CQSH (or GCQSH) equation in the rest of thisstudy. Then the CCHS, real quintic Swift–Hohenberg (RQSH) and real cubic Swift–Hohenberg (RCSH) equationsare particular cases ofEq. (4)with some of the coefficients being equal to 0.

The interpretation of the variables depends on the particular problem. In optics,t is the propagation distance orthe cavity round-trip number (treated as a continuous variable),x is the transverse variable,β1 is the second-orderdiffraction, γ1 is the fourth-order diffraction,µ2 is a nonlinear gain (or 2-photon absorption if negative) andδ

represents a difference between linear gain and loss. The angular spectral gain is represented by the coefficientsβ2

andγ2.Sakaguchi and Brand[23] observed some soliton-like structures in their numerical work. They also showed that

the CQSH equation admits a stable hole solution. Their results indicate that exact solutions may exist, but till nownone have been found. In this paper, we study the CQSH and CCSH equations by using Painlevé analysis and theHirotamulti-linear method. Our method is an extension of the modified Hirota method used earlier by Nozaki andBekki [25] and the modification of the Berloff–Howard method[26]. We note that Nozaki and Bekki solved theCGLE by using the Hirota bilinear method, while Berloff and Howard solved some real-coefficient non-integrableequations using the Weiss–Tabor–Carnevale (WTC) method and the Hirota multi-linear method. In addition, weconfirm that our solutions are valid by using a direct ansatz approach. The solutions we obtain here can be consideredas a basis for further development.

The rest of the paper is organized as follows. The methodology and analytical procedures are described inSection 2.Painlevé analysis of the CQSH and CCSH equations is performed inSection 3. Exact solutions of the generalizedCQSH and CCSH equations and particular cases of them are obtained inSection 4. Finally, we summarize with ourconclusions inSection 5.


2. Methodology

2.1. Painlevé analysis and Hirota multi-linear method

Any solution of an equation must be in accord with the singularity structure of that equation. Painlevé analysis[27]is a tool for investigating that structure. This analysis can be applied both for ordinary differential equations (ODEs)and partial differential equations (PDEs). The power of the Painlevé test lies in its easy algorithmic implementability.The main requirement is the representation of any possible solution in the form of a Laurent expansion in theneighborhood of a movable singularity:

u = F−r∞∑j=0

ujFj , (5)

wherer is the leading-order exponent,F a singularity manifold given byF(z), anduj a set of analytic functionsof z.

There are two necessary conditions for an ODE to pass the Painlevé test:

• the leading-orderr must be an integer, and• it must be possible to solve the recursion relation for the coefficientsuj (z) consistently to any order.

The general expansion of a non-integrable equation will fail the Painlevé test at one of these two steps. Leading-order analysis can be done by balancing the highest-order derivative inx with the strongest nonlinearity.

Weiss et al.[28] developed the singular manifold method to introduce the Painlevé property into the theory ofPDEs. A PDE is said to possess the Painlevé property if its solutions are single-valued about the movable singularitymanifold. To be more specific, if the singularity manifold is given byF(z1, z2, . . . , zn) = 0, then a solution of thePDE must have an expansion of the form ofEq. (5). Substitution of this expansion into the PDE determines thepositive value ofr (from leading-order analysis) and defines the recursion relation for theuj .

Weiss et al.[28] truncated the expansion at the “constant term” level, i.e.

u = u0F−r + u1F

−r+1 + · · · + um−1F−1 + um. (6)

Substituting back into the PDE, one obtains an over-determined system of equations forF anduj . The benefit of thesingular manifold method is that this expansion for a nonlinear PDE contains a lot of information about the PDE.

Most nonlinear non-integrable equations do not possess the Painlevé property, i.e., they are not free of movablecritical singularities[29–32]. For some equations it is still possible to obtain single-valued expansions by putting aconstraint on the arbitrary function in the Painlevé expansion. Such equations are said to be ‘partially integrable’ sys-tems, as presented by Hietarinta[33] as a generalization of the Hirota bilinear formalism for non-integrable systems.He conjectured that all completely integrable PDEs can be put into a bilinear form. There are also non-integrableequations that can be put into the bilinear form and then the partial integrability is associated with the levels ofintegrability defined by the number of solitons that can be combined in anN -soliton solution. Partial integrabilitymeans that the equation allows a restricted number of multisoliton solutions. Berloff and Howard[26] suggestedcombining this treatment of the partial integrability with the use of the Painlevé expansion, truncated before theconstant term level, as a transform to reduce a non-integrable PDE to a multi-linear equation. The Berloff–Howardmethod is a powerful tool for solving non-integrable equations. In this section, we give an example to show how toobtain solutions using this method.

Firstly, we consider the RCSH equation

ut + αuxx + βuxxxx + γ u− δu3 = 0 (7)


and show how to find exact solutions of non-integrable dissipative PDEs by using the Painlevé test and the Hirotamulti-linear method.

We take the transform truncated at the term before the constant term

u = u0F−r + u1F

−r+1 + · · · + um−1F−1. (8)

Analysis of the leading-order terms giver = 2. By substituting this expansion(8) into the RCSHequation (7)andequating the coefficients of the highest powers ofF to 0, we obtain expressions foru0, u1 in terms ofF , and theselead to the transform

u =√

120β/δF 2x

F 2−

√120β/δFxx

F= −

√120β

δ

d2

dx2logF. (9)

This transform leads to an equation which is quadrilinear inF , meaning that each term contains a product of fourfunctions involvingF and its derivatives.

γF 2F 2x − 2FFtF

2x + 6αF 4

x + 2F 2FtFxt − γF 3Fxx + F 2FtFxx − 12αFF2xFxx + 3αF 2F 2

xx − 90βF 2x F

2xx

+90βFF3xx − F 3Fxxt + 4αF 2FxFxxx + 120βF 3

x Fxxx − 120βFFxFxxFxxx + 10βF 2F 2xxx − αF 3Fxxxx

−30βFF2xFxxxx + 15βF 2FxxFxxxx + 6βF 2FxFxxxxx − βF 3Fxxxxxx = 0. (10)

SubstitutingF = 1+ e2kx+2ωt into Eq. (10)and equating the coefficients of different powers of e to 0, we arrive atthe one-soliton solution

u(x) = ±√30γ /δ e2kx

(1 + e2kx)2= ±1

4

√30γ

δsech2(kx), (11)

where

k = ±4√γ

2√

2 4√β

(12)

and we requireα = −5√βγ /2 as a constraint on the equation parameters. This solution is reminiscent of the

sech2-solution, representing an ultrashort soliton at the minimum-dispersion wavelength, taking into account theeffects of fourth-order dispersion[20,34,35]of the equation

i∂ψ

∂t+ α

∂2ψ

∂x2+ β

∂4ψ

∂x4− δ|ψ |2ψ = 0. (13)

Actually, the simple ansatzψ = u(x)exp(−iγ t) reduces this equation to the stationary RCSHequation (7).Now we consider the RQSH equation in the form[36]

ut + αuxx + βuxxxx + γ u− δu3 − ηu5 = 0. (14)

Analysis of the leading-order terms give usr = 1. By substitutingEq. (8)into the RQSHequation (14)and equatingthe coefficients of the highest powers ofF to 0, one obtains expressions foru0, u1 in terms ofF that lead to thetransform

u =4√

24β4√η

Fx

F. (15)


This transform again leads to a quadrilinear equation inF

FtxF3 − FtFxF

2 + βFxxxxxF3 − 5βFxxxxFxF

2 − 10βFxxxFxxF2 + 20βFxxxF

2x F + αFxxxF

3

+30βF 2xxFxF − 60βFxxF

3x − 3αFxxFxF

2 − 2

√6β

ηδF 3

x F + 2αF 3x F + γFxF

3 = 0. (16)

On substitutingF = 1 + e2kx+2ωt into the quadrilinearequation (16)and equating the coefficients of differentpowers of e to 0, we obtain four algebraic equations, but these algebraic equations do not have solutions.

We suppose a different transform

u = G

F. (17)

This leads to an equation which is pentalinear inF andG

F 4G− FxF3G− βFxxxxF

3G+ 8βFxxxFxF2G− 4βFxxxF

3Gx + 6βF 2xxF

2G− 36βFxxF2x FG

+24βFxxFxF2Gx − 6βFxxF

3Gxx − αFxxF3G+ 24βF 4

x G− 24βF 3x FGx + 12βF 2

x F2Gxx + 2αF 2

x F2G

−4βFxF3Gxxx − 2αFxF

3Gx + F 4Gt + βF 4Gxxxx + αF 4Gxx − δF 2G3 − γ ηG5 = 0. (18)

SubstitutingF = 1 + e2kx+2ωt andG = 2u0 ekx+ωt into Eq. (18)and equating the coefficients of different powersof e to 0, we get the one-soliton solution

u = u0 sech(kx), (19)

where

k = ± u0 4√η

4√

24β, u2

0 = 2

3

(− δ

η±√δ2 + 6γ η

η

), (20)

α = −24√

6βγ − √6βηu4

0

12√ηu2

0

. (21)

SubstitutingF = 1 + e2kx+2ωt andG = u0(−1 + e2kx+2ωt ) into Eq. (18)and equating the coefficients of differentpowers of e to 0, we obtain the kink solution

u = u0 tanh(kx), (22)

where

k = ± u0 4√η

4√

24β, u2

0 = − δ

2η± δ2 + 4γ η

2η, α = 3

√6βγ + 2

√6βηu4

0

3√ηu2

0

. (23)

Generally speaking, the application of this technique to complex variable equations is difficult because of thecomplex leading order. However, we will show that this method works in some cases such as the CCSH equation.

2.2. Direct ansatz method

It is well-known that solutions of many nonlinear differential equations can be expressed in terms of hyperbolicfunctions like tanh or sech. This fact motivates the direct method of solution in which the starting point is a suitableansatz so that the p.d.e. is expressible as a polynomial in terms of tanh or sech functions. Clearly this method is


not as general as the Hirota method, since it will not work if there is no solution of the assumed form. However, inprinciple, the method is more straightforward than the Hirota method, and it is more useful in obtaining periodicsolutions (in terms of elliptic functions). In this section, we give an outline of the direct ansatz method and illustratefacets of its application by considering some straightforward examples.

First, we consider the RCSHequation (7)and show how to get exact solutions of non-integrable dissipative PDEsby using the direct ansatz method. The analysis of the leading-order terms give usr = 2, and so we suppose thesolutionu = u0 sech2(kx). We obtain the soliton solution by substituting this ansatz into the RCSH equation. Thisyields an algebraic equation in terms of tanh-functions by using formulae for hyperbolic functions (seeAppendix A),and we then equate the coefficients of different powers of the tanh function to 0. The solution is identical to thatfound by using the Hirota method. We find that the constraint needed on the equation parameters isγ = 4α2/25β.Assuming this is satisfied, the solution is

u(x) = ±α√

3

10βδsech2(kx), (24)

where

k2 = − α

20β. (25)

Now we consider the RQSHequation (14). Analysis of the leading-order terms give usr = 1, and so we supposethe solutionu = u0 sech(kx). Substitution of this ansatz into the RQSH equation yields an algebraic equation interms of the tanh function. Again, by equating the coefficients of different powers of the tanh function to 0, weobtain the soliton solution, and note that it is identical with the Hirota method solution.

We find that the constraint needed on the equation parameters is now

− 9α2

100β+ γ + 3δ2

50η± 2

√6αδ

25√βη

= 0. (26)

With this satisfied, we have

u = u0 sech(kx), (27)

where

k = ± u0 4√η

4√

24β, u2

0 = −6√βδ ± √

6ηα

5√βη

. (28)

In the same way, we obtain a kink solution by the ansatzu0 tanh(kx)

u = u0 tanh(kx), (29)

where

k = ± u0 4√η

4√

24β, u2

0 = − δ

2η± δ2 + 4γ η

2η(30)

and the constraint needed on the equation parameters is

α = 3√

6βγ + 2√

6βηu40

3√ηu2

0

. (31)

It is easy to develop this method to get solutions in terms of elliptic functions. If we substitute a Jacobi ellipticfunction, as an ansatz, into the RQSHequation (14), we obtain an algebraic equation in terms of some Jacobi elliptic


function. By using the formulae inAppendix A, and equating the coefficients of different powers of the ellipticfunction to 0, we then get the elliptic solutions.

The elliptic function solutions of the RQSHequation (14)are as follows:

• Jacobi cn function

u = u0 cn(kx, q), (32)

where

k = ±√α − 2αq ±

√(2q − 1)2α2 − 4βγA

2βA, (33)

A = 16q2 − 16q + 1, (34)

u20 = −2k2q(α + 10β(2q − 1)k2)

δ(35)


η = 24βq2k4

u40

. (36)

• Jacobi sn function

u = u0 sn(kx, q), (37)

where

k = ±√α + αq ±

√(q + 1)2α2 − 4βγA

2βA, (38)

A = q2 + 14q + 1, (39)

u20 = 2k2q(α − 10β(q + 1)k2)

δ(40)


η = 24βq2k4

u40

. (41)

• Jacobi dn function

u = u0 dn(kx, q), (42)

where

k = ±√(q − 2)α ±

√(q − 2)2α2 − 4βγA

2βA, (43)

A = q2 − 16q + 16, (44)

u20 = 2k2(−α + 10(q − 2)βk2)

δ(45)



η = 24βk4

u40

. (46)

In principle, these Jacobi elliptic function solutions can be also obtained from multi-linear forms, because eachJacobi elliptic function can be expressed by theta functions which are Hirotaτ -functions in multi-linear form[37,38]. However, calculation by this method is tedious, so we do not provide details of it here.

Next we consider an elliptic function solution of the RCSH equation. We know from Painlevé analysis that theRCSH equation has a double pole. Thus we look for elliptic functions having double poles. We suppose

u = u0 + ℘(kx). (47)

Substituting this ansatz into the RCSHequation (7)yields an algebraic equation in terms of a Weierstrass℘ function.By using formulae fromAppendix A, and equating the coefficients of different powers of the Weierstrass℘ functionto 0, we obtain the following elliptic function solution:

u = u0 + ℘(kx), (48)

where

k = ±(

δ

120β

)1/4

, u0 = ± α√30βδ

, (49)

g2 = 2(10βγ − α2)

3βδ, g3 = ±2

√2α(α2 − 5βγ )

3√

15β3δ3. (50)

From its relation to the Weierstrassσ function (seeAppendix A), we know that the Weierstrassσ function is aτ -function of the Hirota multi-linear form. Thus, we can construct this solution by the Hirota multi-linear method.However, we do not give the detail of this approach here.

We can find another elliptic function solution of the RCSH equation

u = u0 + cn2(kx, q), (51)

where

k = ±(

δ

120βq2

)1/4

, (52)

q = 6 + 19u0 + 15u20 ± √

A

8 + 38u0 + 60u20 + 30u3

0

, (53)

A = −15u40 − 30u3

0 − 3u20 + 12u0 + 4 (54)

and the constraints needed on the equation parameters are

α = 20β(1 − (2 + 3u0)q), (55)

γ = 8βk4(8 − (23+ 30u0)q + (23+ 60u0 + 45u20)q

2). (56)

We note that all above hyperbolic-function solutions can be derived in the limit ofq → 1 of above elliptic functionsolutions.


3. Painlevé analysis of the CCSH equation

It is difficult to use a full expansion because the CQSH equation possesses both a complex leading order andnon-integer resonances, and it generally has no consistency conditions. To see this, the CQSH equation is firstrewritten as the following system of equations:

iψt + (β1 + iβ2)ψxx + (γ1 + iγ2)ψxxxx + (µ1 + iµ2)|ψ |2ψ + (ν1 + iν2)|ψ |4ψ = iδψ, (57)

−iψ∗t + (β1 − iβ2)ψ

∗xx + (γ1 − iγ2)ψ

∗xxxx + (µ1 − iµ2)|ψ |2ψ∗ + (ν1 − iν2)|ψ |4ψ∗ = −iδψ∗. (58)

To leading order, we setψ andψ∗ as

ψ ∼ ψ0F−ξ1, ψ∗ ∼ ψ∗

0F−ξ2, (59)

whereξ1 andξ2 are the leading-order exponents. Upon equating the exponents of the dominant balance terms inCQSH

(γ1 + iγ2)ψxxxx + (ν1 + iν2)|ψ |4ψ ∼ 0, (60)

we find

ξ1 + ξ2 = 2. (61)

To find ξ1 andξ2 we must also equate the coefficients in front of these terms. FromEq. (57)we get

|ψ0|4 = −γ1 + iγ2

ν1 + iν2ξ1(ξ1 + 1)(ξ1 + 2)(ξ1 + 3)

(∂F

∂x

)4

(62)

and fromEq. (58)

|ψ0|4 = −γ1 − iγ2

ν1 − iν2ξ2(ξ2 + 1)(ξ2 + 2)(ξ2 + 3)

(∂F

∂x

)4

. (63)

Eqs. (62) and (63)are combined to give

γ1 + iγ2

ν1 + iν2ξ1(ξ1 + 1)(ξ1 + 2)(ξ1 + 3) = γ1 − iγ2

ν1 − iν2ξ2(ξ2 + 1)(ξ2 + 2)(ξ2 + 3). (64)

Eqs. (61) and (64)are now the two equations we need to solve forξ1 andξ2. The result is

ξ1 = 1 + iα, ξ2 = 1 − iα, (65)

whereα satisfies

(α4 − 35α2 + 24)(γ1ν2 − γ2ν1)+ 10α(5 − α2)(γ1ν1 + γ2ν2) = 0. (66)

(To haveα = 0, we needγ1ν2 = γ2ν1 as the condition for a chirp-less solution.) The leading-order exponentsξ1 = 1 + iα andξ2 = 1 − iα are not integers unlessα = 0. This means that the CQSH equation already fails thePainlevé test at the first step ifα = 0.

Whenγ1ν2 − γ2ν1 = 0 we can haveα = 0 (orα = √5). In fact this is the condition of the next section, giving

the chirp-less solution. This plain solution is not possible for the quintic CGL equation.The expansions forψ andψ∗ therefore take the form

ψ = (ψ0F−1−iα + ψ1F

−iα + · · · )exp(iKx + iΩt), (67)

ψ∗ = (ψ∗0F

−1+iα + ψ1Fiα + · · · )exp(−iKx − iΩt). (68)


The resonances for the CQSH equation can be calculated from the recursion relations for theψj ’s andψ∗j ’s, but

we do not include them here because they are not important for our purpose.We can expect the following dependent variable transformation by the above analysis

ψ = G

F 1+iαexp(iΩt), ψ∗ = G∗

F 1−iαexp(−iΩt). (69)

(In general, the transform should beψ = (G/F 1+iα)exp(iKx+ iΩt), however, in our case, this transformation alsogives solutions in the next section.)

In the same way, we can obtain the dependent variable transformation of the CCSH equation

ψ = G2

F 2+iαexp(iΩt), ψ∗ = G∗2

F 2−iαexp(−iΩt). (70)

In the next section, we show the existence of exact solutions by using these transformations.

4. Exact solutions of the complex S–H equation

4.1. The CQSH equation

We consider the CQSH equation in[23]

iψt + (f + 2id)ψxx + idψxxxx + (b2 − ib1)|ψ |2ψ + (−c2 + ic1)|ψ |4ψ = (−a2 + i(a1 − d))ψ, (71)

(In the numerical work of Sakaguchi and Brand,a2 was 0.) Substituting the transformation(69) into the CQSHequation (71), we obtain a pentalinear equation

(id −Ωa1 + a2 − i)F 4G+ (b2 − ib1)F2G2G∗ − (c2 − ic1)G

3G∗2 − iF 3FtG+ iF 4Gt

+(2f + 4id)F 2F 2x G+ 24idF4

xG− (2f + 4id)F 3FxGx − 24idFF3xGx − (2id + f )F 3FxxG

−36idFF2xFxxG+ 24idF2FxFxxGx + 6idF2F 2

xxG+ (f + 2id)F 4Gxx + 12idF2F 2x Gxx − 6idF3FxxGxx

+8idF2FxFxxxG− 4idF3FxxxGx − 4idF3FxGxxx − idF3FxxxxG+ idF4Gxxxx = 0. (72)

PuttingF , G andG∗ as polynomials in terms of exp(kx + ωt) and substituting these functions intoEq. (72)andequating the coefficients of different powers of e to 0, we obtain the following exact solutions. We can also obtainsame solutions by using direct ansatz method.

• Bright soliton

ψ = g sech(kx)eiΩt , (73)

where

k = ±√a1 − √

d√d

, Ω = −f + f√a1√d

+ a2, |g|2 = 2f (√a1 − √

d)√db2

. (74)

The coefficients must satisfy the following relations:

b1 = b22(4d − 5

√da1)

f, c1 = −6b2

2d

f 2, c2 = 0. (75)


• Dark soliton

ψ = g tanh(kx)eiΩt , (76)

where

k = ±√

fb1 + 2db2

20db2, Ω = −2fk2 + a2, |g|2 = −2fk2

b2. (77)


a1 = d(16k4 − 4k2 + 1), c1 = −6db22

f 2, c2 = 0. (78)

• Chirped bright soliton

ψ = g sech(kx)e−iα log(sech(kx)) eiΩt , (79)

where

k = ±√f α + d(α2 − 1)±

√d(d − a1)A+ (f α + d(α2 − 1))2

dA, (80)

A = α4 − 6α2 + 1, (81)

Ω = k2(f − f α2 + 4dα + 4dα(1 − α2)k2)+ a2, (82)

|g|2 = k2(2dα(23− 7α2)k2 + 6dα − f (α2 − 2))

b2. (83)


b1 = k2(3f α − 2d(2 − α2)− 2d(10− 19α2 + α4)k2)

|g|2 , (84)

c1 = −b22

d(α4 − 35α2 + 24)

(2dα(23− 7α2)k2 + 6dα − f (α2 − 2))2, (85)

c2 = −b22

10dα(α2 − 5)

(2dα(23− 7α2)k2 + 6dα − f (α2 − 2))2. (86)

• Chirped dark soliton

ψ = g tanh(kx)e−iα log(sech(kx)) eiΩt , (87)

where

k = ±√

2

A− 3f α

2dA±√(4d − 3f α)2 − 4d(d − a1)A

2dA, (88)

A = 16− 15α2, (89)

Ω = k2((−2f − 6dα)+ 30dαk2)+ a2, (90)

|g|2 = −k2(2f + 6dα − f α2 + 10dα(α2 − 8)k2)

b2. (91)



b1 = −k2(−4d + 3f α + 2dα2 + 10d(4 − 5α2)k2)

|g|2 , (92)

c1 = −dk4(α4 − 35α2 + 24)

|g|4 , c2 = −10dαk4(α2 − 5)

|g|4 . (93)

Elliptic function solutions of the CQSH equation are the following:

• Jacobi cn function solution

ψ = g cn(kx, q)eiΩt , (94)

where

k = ±√d(1 − 2q)±√

d(−12dq(q − 1)+ a1A)

dA, (95)

A = 16q2 − 16q + 1, (96)

Ω = fk2(2q − 1)+ a2, |g|2 = 2fqk2

b2. (97)


b1 = −4dqk2(1 + 5(2q − 1)k2)

|g|2 , c1 = −24dq2k4

|g|4 , c2 = 0. (98)

• Jacobi sn function solution

ψ = g sn(kx, q)eiΩt , (99)

where

k = ±√d(q + 1)±√

d(−12dq + a1A)

dA, (100)

A = q2 + 14q + 1, (101)

Ω = −fk2(q + 1)+ a2, |g|2 = −2fqk2

b2. (102)


b1 = −4dqk2(−1 + 5(q + 1)k2)

|g|2 , c1 = −24dq2k4

|g|4 , c2 = 0. (103)

• Jacobi dn function solution

ψ = g dn(kx, q)eiΩt , (104)

where

k = ±√d(q − 2)± √

d(12d(q − 1)+ a1A)

dA, (105)


A = q2 − 16q + 16, (106)

Ω = −fk2(q − 2)+ a2, |g|2 = 2fk2

b2. (107)


b1 = 4dk2(−1 + 5(q − 2)k2)

|g|2 , c1 = −24dk4

|g|4 , c2 = 0. (108)

4.2. The generalized complex quintic Swift-Hohenberg (GCQSH) equation

Now we consider the GCQSH equation

iψt + βψxx + γψxxxx + µ|ψ |2ψ + ν|ψ |4ψ = iδψ, (109)

where all coefficientsβ = β1 + iβ2, γ = γ1 + iγ2, µ = µ1 + iµ2 andν = ν1 + iν2 are complex andδ is real.This equation can be easily normalized by rescalingt ′ = µ1t , x′ = √

mu1/β1x, so thatβ1 andµ1 can be 1 ifβ1

andµ1 are non-zero.Substituting the transformation(69) into the GCQSHequation (109), we obtain a pentalinear equation

νG3G∗2 + (24+ 50iα − 35α2 − 10iα3 + α4)γF 4x G+ 2(−6 − 11iα + 6α2 + iα3)γ

×(2FF3xGx + 3FF2

xFxxG)+ µF 2G2G∗ − 6(−2 − 3iα + α2)γ (2F 2FxFxxGx + F 2F 2x Gxx)

−(−2 − 3iα + α2)(βF 2F 2x G+ 3γF 2F 2

xxG+ 4γF 2FxFxxxG)− i(−i + α)(2βF 3FxGx

+6γF 3FxxGxx + 4γF 3FxxxGx + 4γF 3FxGxxx + iF 3FtG+ βF 3FxxG

+γF 3FxxxxG)+ (−iδ −Ω)F 4G+ iF 4Gt + βF 4Gxx + γF 4Gxxxx. (110)

PuttingF , G andG∗ as polynomials in terms of exp(kx + ωt) and substituting these functions into pentalinearequation (110)and equating the coefficients of different powers of e to 0, we get the following solutions. We canalso obtain same solutions by using direct ansatz method.

First, we consider chirp-less (α = 0) solutions. FromEq. (66)we needγ1ν2 = γ2ν1.

• Bright soliton

ψ = g sech(kx)eiΩt , (111)

where

k = ±

√√√√−β2 ±√β2

2 + 4δγ2

2γ2, (112)

Ω = k2(1 + γ1k2), |g|2 = 2k2(1 + 10γ1k

2). (113)


µ2 = 2k2(β2 + 10γ2k2)

|g|2 , ν1 = −24γ1k4

|g|4 , ν2 = −24γ2k4

|g|4 , (114)

andβ1 = µ1 = 1 (normalized coefficients).


• Dark soliton

ψ = g tanh(kx)eiΩt , (115)

where

k = ±1

4

√√√√β2 ±√β2

2 + 16δγ2

γ2, (116)

Ω = 2k2(8γ1k2 − 1), |g|2 = 2k2(20γ1k

2 − 1). (117)


µ2 = 2k2(20γ2k2 − β2)

|g|2 , ν1 = −24γ1k4

|g|4 , ν2 = −24γ2k4

|g|4 (118)

andβ1 = µ1 = 1 (normalized coefficients).• Chirped bright soliton

ψ = g sech(kx)e−iα log(sech(kx)) eiΩt , (119)

where

k = ±√

2α + (α2 − 1)β2 ±√(2α + (α2 − 1)β2)2 + 4δA

2A, (120)

A = 4α(α2 − 1)γ1 + (α4 − 6α2 + 1)γ2, (121)

Ω = k2((1 − α2)+ 2αβ2)+ k4((1 − 6α2 + α4)γ1 + (4α − 4α3)γ2), (122)

|g|2 = k2((2 − α2)+ 3αβ2)+ 2k4((10− 19α2 + α4)γ1 + (23α − 7α3)γ2). (123)


µ2 = k2((2 − α2)β2 − 3α)

|g|2 + 2k4((10− 19α2 + α4)γ2 − (23α + 7α3)γ1)

|g|2 , (124)

ν1 = −k4((24− 35α2 + α4)γ1 − 10α(α2 − 5)γ2)

|g|4 , (125)

ν2 = −k4((24− 35α2 + α4)γ2 + 10α(α2 − 5)γ1)

|g|4 , (126)

andβ1 = µ1 = 1 (normalized coefficients).• Chirped dark soliton

ψ = g tanh(kx)e−iα log(sech(kx)) eiΩt , (127)

where

k = ±√

3α − 2β2 ±√(3α − 2β2)2 − 4δA

2A, (128)

A = 30αγ1 + (15α2 − 16)γ2, (129)


Ω = k2(−2 − 3αβ2 + (16γ1 − 15α2γ1 + 30αγ2)k2), (130)

|g|2 = k2((α2 − 2)− 3αβ2 + 10(4γ1 − 5α2γ1 + 8αγ2 − α3γ2)k2). (131)


µ2 = k2((α2 − 2)β2 + 3α + 10(4γ2 − 5α2γ2 − 8αγ1 + α3γ1)k2)

|g|2 , (132)

ν1 = −k4((24− 35α2 + α4)γ1 − 10α(α2 − 5)γ2)

|g|4 , (133)

ν2 = −k4((24− 35α2 + α4)γ2 + 10α(α2 − 5)γ1)

|g|4 (134)

andβ1 = µ1 = 1 (normalized coefficients).Elliptic function solutions of the GCQSH equation are the following:

• Jacobi cn function solution

ψ = g cn(kx, q)eiΩt , (135)

where

k = ±

√√√√ (1 − 2q)β2 ±√(1 − 2q)2β2

2 + 4δγ2A

2γ2A, (136)

Ω = k2((2q − 1)+ γ1Ak2), (137)

A = 16q2 − 16q + 1, (138)

|g|2 = 2qk2(1 + 10γ1(2q − 1)k2). (139)


µ2 = 2qk2(β2 + 10γ2(2q − 1)k2)

|g|2 , (140)

ν1 = −24γ1q2k4

|g|4 , ν2 = −24γ2q2k4

|g|4 , (141)

andβ1 = µ1 = 1 (normalized coefficients).• Jacobi sn function solution

ψ = g sn(kx, q)eiΩt , (142)

where

k = ±

√√√√ (1 + q)β2 ±√(q + 1)2β2

2 + 4δγ2A

2γ2A, (143)

Ω = k2(−(q + 1)+ γ1Ak2), (144)


A = q2 + 14q + 1, (145)

|g|2 = 2qk2(−1 + 10γ1(q + 1)k2). (146)


µ2 = 2qk2(−β2 + 10γ2(q + 1)k2)

|g|2 , (147)

ν1 = −24γ1q2k4

|g|4 , ν2 = −24γ2q2k4

|g|4 , (148)

andβ1 = µ1 = 1 (normalized coefficients).• Jacobi dn function solution

ψ = g dn(kx, q)eiΩt , (149)

where

k = ±

√√√√ (q − 2)β2 ±√(q − 2)2β2

2 + 4δγ2A

2γ2A, (150)

Ω = k2(−(q − 2)+ γ1Ak2), (151)

A = q2 − 16q + 16, (152)

|g|2 = 2k2(1 − 10γ1(q − 2)k2). (153)


µ2 = 2k2(β2 − 10γ2(q − 2)k2)

|g|2 , (154)

ν1 = −24γ1k4

|g|4 , ν2 = −24γ2k4

|g|4 , (155)


4.3. The CCSH equation

Now we consider the CCSH equation

ψt = (µ+ iν)ψ + ifψxx − d(ψ + 2ψxx + ψxxxx)− (δ + iγ )|ψ |2ψ. (156)

Substituting the transformation(70) into the CCSHequation (156), we obtain a hexalinear equation

(−iγ − δ)G4G∗2 − d(120+ 154iα − 71α2 − 14iα3 + α4)F 4x G

2 + 2d(24+ 26iα − 9α2 − iα3)

×(4FF3xGGx + 3FF2

xFxxG2)+ (α2 − 5iα − 6)(12dF2F 2

x G2x + 12d(2F 2FxFxxGGx + F 2F 2

x GGxx)

+(2dF2F 2x G

2 − ifF2F 2x G

2 + 3dF2F 2xxG

2 + 4dF2FxFxxxG2))+ (2 + iα)(12dF3FxxG

2x + 24dF3FxGxGxx

+8dF3FxGGx − 4ifF3FxGGx + 12dF3FxxGGxx + 8dF3FxxxGGx + 8dF3FxGGxxx + F 3FtG2

+2dF3FxxG2 − ifF3FxxG

2 + dF3FxxxxG2)+ 2ifF4G2

x − (d − µ− iν + iΩ)F 4G2 − 4dF4G2x − 6dF4G2

xx

−8dF4GxGxxx − 2F 4GGt − 4dF4GGxx + 2ifF4GGxx − 2dF4GGxxxx. (157)


PuttingF , G andG∗ as polynomials in terms of exp(kx + ωt) and substituting these functions into hexalinearequation (157)and equating the coefficients of different powers of e to 0, we get the following solutions. We canalso obtain same solutions by using direct ansatz method.

• Chirped bright soliton

ψ = g sech2(kx)e−α log(sech(kx)) eiΩt , (158)

k = ± 1√α2 − 10

, |g|2 = −(120− 71α2 + α4)dk4, (159)

µ = 4(9 + 8α2)dk4, f = −12αdk2, (160)

Ω = ν − αdk4(96− 12α2), γ = (154α − 14α3)dk4

|g|2 . (161)

Now we look for solutions of elliptic function. We suppose

ψ = (g + ℘(kx))eiΩt . (162)

Substituting this ansatz into the CCSHequation (156)yields an algebraic equation in terms of a Weierstrass℘

function by using formulas inAppendix A, and equating the coefficients of different powers of a Weierstrass℘

function to 0, we obtain the following elliptic function solution

ψ = (g + ℘(kx))eiΩt , (163)

where

k = ±( −δ

120d

)1/4

, g = ±√

− 2d

15δ, Ω = ν, (164)

g2 = 4d(5µ− 3d)

3δ, g3 = ±4(d − 5µ)

3

√−2d

15δ3, (165)

f = 0, γ = 0. (166)

4.4. The GCCSH equation

We consider the GCCSH equation

iψt + βψxx + γψxxxx + µ|ψ |2ψ = δψ, (167)

where all coefficientsβ = β1 + iβ2, γ = γ1 + iγ2, µ = µ1 + iµ2 andδ = δ1 + iδ2 are complex.This equation can be easily normalized by rescalingt ′ = µ1t , x′ = √

mu1/β1x, so thatβ1 andµ1 can be 1 ifβ1

andµ1 are non-zero.Substituting the transformation(70) into the GCCSHequation (167), we obtain a hexalinear equation

µG∗2G4 + (120+ 154iα − 71α2 − 14iα3 + α4)γF 4

x G2 + 2(−24− 26iα + 9α2 + iα3)γ

×(4FGF2xFxGx + 3FGF2

xGFxx)− (−6 − 5iα + α2)(12γF 2F 2x G

2x + 24γF 2FxGGxFxx

+12γF 2F 2x GGxx + βF 2F 2

x G2 + 3γF 2F 2

xxG2 + 4γF 2FxFxxxG

2)− i(−2i + α)F 3

×(12γ (G2xFxx + 2FxGxGxx)+ 4(βFxGGx + 3γFxxGGxx + 2γFxxxGGx + 2γFxGGxxx)

+iFtG2 + βFxxG

2 + γFxxxxG2)+ (−iδ −Ω)F 4G2 + 2(βF 4G2

x + 3γF 4G2xx + 4γF 4GxGxxx)

+2(iF 4GGt + βF 4GGxx + γF 4GGxxxx). (168)


PuttingF,G andG∗ as polynomials in terms of exp(kx + ωt) and substituting these functions into hexalinearequation (168)and equating the coefficients of different powers of e to 0, we get the following solutions. We canalso obtain same solutions by using direct ansatz method.

• Bright soliton

ψ = g sech2(kx)eiΩt , (169)

where

k = ±1

4

√5δ2

β2, Ω = 16k2

5− δ1, (170)

|g|2 = 6k2, µ2 = β2, γ1 = − 4β2

25δ2, γ2 = − 4β2

2

25δ2. (171)

andβ1 = µ1 = 1 (normalized coefficients).• Chirped bright soliton

ψ = g sech2(kx)e−iα log(sech(kx)) eiΩt , (172)

where

k = ±√

−2(α2 + 2α + 10)(α2 − 2α + 10)δ2

6α(α4 + 16α2 + 96)β1 + β2A, (173)

A = α6 + 10α4 + 8α2 − 640, (174)

Ω = k2(−(α6 + 10α4 + 8α2 − 640)+ 6αβ2(α4 + 16α2 + 96))

2(α2 + 2α + 10)(α2 − 2α + 10)− δ1, (175)

|g|2 = k2((−α6 − 3α4 + 94α2 + 1200)+ 4α(2α4 + 33α2 + 205)β2)

2(α2 + 2α + 10)(α2 − 2α + 10). (176)


µ2 = k2((−α6 − 3α4 + 94α2 + 1200)β2 − 4α(2α4 + 33α2 + 205))

2|g|2(α2 + 2α + 10)(α2 − 2α + 10), (177)

γ1 = (α2 − 10)+ 6αβ2

2k2(α2 + 2α + 10)(α2 − 2α + 10), (178)

γ2 = (α2 − 10)β2 + 6α

2k2(α2 + 2α + 10)(α2 − 2α + 10)(179)


Now we look for solutions of elliptic function. We suppose

ψ = (g + ℘(kx))eiΩt . (180)

Substituting this ansatz into the GCCSHequation (167)yields an algebraic equation in terms of a Weierstrass℘

function by using formulas inAppendix A, and equating the coefficients of different powers of a Weierstrass℘

function to 0, we obtain the following elliptic function solution:

ψ = (g + ℘(kx))eiΩt , (181)


where

k = ±( −1

120γ1

)1/4

, g = ± 1√−30γ1, Ω = −2 + γ1(3g2 − 20δ1)

20γ1, (182)

g2 = 2β22 + 20γ2δ2

3γ2δ2, g3 = ±2

√30(1 + 5γ1(Ω + δ1))

45√

−γ 31

. (183)


µ2 = −120γ2k4, β2

2 = −30γ2µ2g2, (184)

135γ 32µ

32g

23 = −8β2

2(β22 + 5γ2δ2)

2 (185)

andβ1 = µ1 = 1 (normalized coefficients).We find another elliptic function solution

ψ = (g + cn2(kx, q))eiΩt , (186)

where

k = ±( −1

120γ1q2

)1/4

, (187)

Ω = −δ1 − 8γ1k4(8 − (23+ 30g)q + (23+ 60g + 45g2)q2), (188)

q = 6 + 19g + 15g2 ± √A

2(g + 1)(15g2 + 15g + 4), (189)

A = −15g4 − 30g3 − 3g2 + 12g + 4, (190)

g = 1 − 6q

3q− 1

60γ1q. (191)


µ2 = −120γ2q2k4, (192)

β2 = 20γ2(1 − (2 + 3g)q), (193)

δ2 = −8γ2k4(8 − (23+ 30g)q + (23+ 60g + 45g2)q2) (194)


5. Conclusion

A number of pattern formation phenomena are described by the (2+ 1)-dimensional complex S–H equation.Among them is an approximation of a Maxwell–Bloch system for the transverse dynamics of a two-level classB laser, filamentation in wide aperture semiconductor lasers, etc. (1+ 1)-dimensional version of this equation isone of the basic equations for modeling temporal behavior of laser systems. Among others, it describes, shortpulse generation by passively mode-locked lasers with complicated spectral filtering elements. The knowledge ofits solutions presented in any form will help to understand better the processes in such systems. In this paper,


using Painlevé analysis, Hirota multi-linear method and direct ansatz technique, we studied analytic solutions ofthe (1+ 1)-dimensional complex cubic and quintic Swift–Hohenberg equations. We considered both, standard andgeneralized versions of these equations. We have found that a number of exact solutions exist to each of theseequations provided that coefficients are bound by special relations. The set of solutions include particular types ofsolitary wave solutions, hole (dark soliton) solutions and periodic solutions in terms of elliptic Jacobi functions andWeierstrass℘ function. Clearly, these solutions represent only a small subset of large variety of possible solutionsadmitted by the complex cubic and quintic Swift–Hohenberg equations. Nevertheless, the solutions presented hereare found for the first time and they might serve as seeding solutions for a wider class of localized structures which,no doubt, exist in these systems. We also hope that they will be useful in further numerical analysis of varioussolutions to the CCSH equation.

Acknowledgements

KM is grateful to H. Sakaguchi and N. Berloff for stimulating discussions and helpful suggestions. KM wassupported by a JSPS Fellowship for Young Scientists. NA acknowledges financial support from the Secretaría deEstado de Educación y Universidades, Spain, Reference No. SAB2000-0197 and support from US AROFE (grantN62649-02-1-0004).

Appendix A. Formulas of hyperbolic functions and elliptic functions

Let Dn = dn(kx, q), Sn= sn(kx, q), Cn= cn(kx, q).Jacobi elliptic functions satisfy the following two relations:

Sn2 + Cn2 = 1, (A.1)

Dn2 = 1 − q Sn2. (A.2)

We have the following differential formulas:

1

k2

Dn′′(x)Dn(x)

= q(−1 + 2 Sn2) (A.3)

and

1

qk4

Dn′′′′(x)Dn(x)

= 4 + q − 4(2 + 5q)Sn2 + 24q Sn4. (A.4)

Now the limitq = 1 reduces this to dn(kx,1) = sech(kx) = S(x), sn(kx,1) = tanh(kx) = T (x). Then

1

k2

S′′(x)S(x)

= −1 + 2T 2 (A.5)

and

1

k4

S′′′′(x)S(x)

= 5 − 28T 2 + 24T 4. (A.6)

The other main function we need for the periodic solutions is Jacobi sn function. We have the following differentialformulas:

1

k2

Sn′′(x)Sn(x)

= 2q Sn2(x)− q − 1, (A.7)


and

1

k4

Sn′′′′(x)Sn(x)

= 1 + 14q + q2 − 20q(q + 1)Sn2(x)+ 24q2 Sn4(x). (A.8)

Now the limitq = 1 reduces this to sn(kx,1) = tanh(kx) = T (x). Then

1

k2

T ′′(x)T (x)

= 2(−1 + T 2), (A.9)

and

1

k4

T ′′′′(x)T (x)

= 8(2 − 5T 2 + 3T 4). (A.10)

We give the following formula of Jacobi cn function:

1

k2

Cn′′(x)Cn(x)

= −1 + 2q Sn2(x) (A.11)

and

1

k4

Cn′′′′(x)Cn(x)

= 1 + 4q − 4q(2q + 5)Sn2(x)+ 24q2 Sn4(x). (A.12)

Finally, we give the following differential formulas of Weierstrass℘ function:

[℘′(z)]2 = 4[℘(z)]2 − g2℘(z)− g3, (A.13)

℘′′(z) = 6[℘(z)]2 − 12g2, (A.14)

℘′′′′(z) = 12(10[℘(z)]3 − 32g2℘(z)− g3), (A.15)

whereg2 andg3 are constants. Weierstrass℘ function is connected to Weierstrassσ function by

℘(x) = − d2

dx2logσ(x). (A.16)

Weierstrass℘ function can be expressed by Jacobi elliptic functions:

℘(x) = e3 + e1 − e3

sn2(kx, q), k2 = e1 − e3, q = e2 − e3

e1 − e3, (A.17)

where

e1 + e2 + e3 = 0, e1e2 + e1e3 + e2e3 = −14g2, e1e2e3 = 1

4g3. (A.18)

References

[1] M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65 (1993) 851.[2] J. Buceta, K. Lindenberg, J.M.R. Parrondo, Phys. Rev. Lett. 88 (2002) 024103.[3] J.B. Swift, P.C. Hohenberg, Phys. Rev. A 15 (1977) 319.[4] M.I. Tribelskii, Sov. Phys.-Uspekhi 40 (1997) 159.[5] P.C. Hohenberg, J. Swift, Phys. Rev. A 46 (1992) 4773.[6] K. Staliunas, V.J. Sanchez-Morcillo, Phys. Lett. A 241 (1998) 28.[7] I. Aranson, M. Assenheimer, V. Steinberg, Phys. Rev. E 55 (1997) R4877.[8] V.J. Sanchez-Morcillo, E. Roldan, G.J. de Valcarcel, Phys. Rev. A 56 (1997) 3237.[9] J. Garcia-Ojalvo, R. Vilaseca, M.C. Torrent, Phys. Rev. A 56 (1997) 5111.


[10] J. Lega, J.V. Moloney, A.C. Newell, Phys. Rev. Lett. 73 (1994) 2978.[11] C.O. Weiss, in: A.D. Boardman, A.P. Sukhorukov (Eds.), Soliton-driven Photonics, Kluwer Academic Publishers, Dordrecht, 2001, p. 169.[12] I. Aranson, D. Hochheiser, J.V. Moloney, Phys. Rev. A 55 (1997) 3173.[13] I. Aranson, L. Kramer, Rev. Mod. Phys. 74 (2002) 99.[14] V.J. Sanchez-Morcillo, K. Staliunas, Phys. Rev. E 60 (1999) 6153.[15] K. Staliunas, Phys. Rev. E 64 (2001) 066129.[16] J. Lega, J.V. Moloney, A.C. Newell, Physica D 83 (1995) 478.[17] N.R. Pereira, L. Stenflo, Phys. Fluids 20 (1977) 1733.[18] W. van Saarloos, P.C. Hohenberg, Physica D 56 (1992) 303.[19] N. Bekki, K. Nozaki, Phys. Lett. A 110 (1985) 133.[20] N. Akhmediev, A. Ankiewicz, Solitons, Nonlinear Pulses and Beams, Chapman & Hall, London, 1997.[21] N. Akhmediev, in: A.D. Boardman, A.P. Sukhorukov (Eds.), Soliton-driven Photonics, 2001, p. 371.[22] N. Akhmediev, A. Ankiewicz, in: S. Trillo, W.E. Toruellas (Eds.), Spatial Solitons 1, Springer, Berlin, 2001, p. 311.[23] H. Sakaguchi, H.R. Brand, Physica D 117 (1998) 95.[24] J.D. Moores, Optics Commun. 96 (1993) 65–70.[25] K. Nozaki, N. Bekki, J. Phys. Soc. Jpn. 53 (1984) 1581.[26] N.G. Berloff, L.N. Howard, Stud. Appl. Math. 99 (1997) 1.[27] M.J. Ablowitz, P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Notes

Series 149, Cambridge University Press, Cambridge, 1991.[28] J. Weiss, M. Tabor, G. Carnevale, J. Math. Phys. 24 (1983) 522.[29] F. Cariello, M. Tabor, Physica D 39 (1989) 77.[30] R. Conte, M. Musette, Physica D 69 (1993) 1.[31] P. Marcq, H. Chaté, R. Conte, Physica D 73 (1994) 305.[32] A. Ankiewicz, N. Akhmediev, P. Winternitz, J. Eng. Math. 36 (1999) 11.[33] J. Hietarinta, in: R. Conte, N. Boccara (Eds.), Partially Integrable Evolution Equations in Physics, Kluwer Academic Publishers, Dordrecht,

1990, p. 459.[34] M. Karlsson, A. Höök, Optics Commun. 104 (1994) 303.[35] N.N. Akhmediev, A.V. Buryak, M. Karlsson, Optics Commun. 110 (1994) 540.[36] H. Sakaguchi, H.R. Brand, Physica D 97 (1996) 274.[37] K.W. Chow, J. Math. Phys. 36 (1995) 4125.[38] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1927.

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